Okay well, to be honest, Turnip HAS no story to him. You dig him up from the ground and after some pointless accented spiel he'll join you. What can be said of him... he's a turnip who fights with a sword. Yay.

Recruitment
Turnip will likely be one of the last characters you'll get. You need to first get the Ice Gun, and head to the Hermit's Hideaway in Home World. On the lower part of the screen, near where you first enter, there's an area on the ground that, if you search it, is warm. Too warm in fact, for anything to grow there. This is where the Ice Gun comes in handy. Use it at that spot, and later on, you'll see a plant sprouting there. Bring Poshul to dig it up, and a large turnip, conveniently named Turnip, will join you.